Some 44 percent of respondents were in support, compared with 32 percent who opposed it and 24 percent who were unsure.
The poll asked simply whether people supported the bill, and presented a brief summary of the arguments both for and against.
"The government has introduced a Fast-track Approvals Bill that it says will speed up the process for consenting infrastructure projects and reduce costs. Opponents of the bill say that it gives too much power to Government ministers and that it does not have sufficient environmental protections. Do you support or oppose the FastTrack Approvals Bill?"
The question was included as part of a wider survey.
Men were far more supportive of the idea, with 55 percent supporting and 28 percent opposed compared to 34 percent of women in support and 35 percent opposed.
Age was also a factor, with 51 percent of those aged 60+ in support, compared to 44 percent of those aged 40 to 59, and 38 percent of those under 40.
People who said they voted for one of the coalition parties were much more likely to say they supported it, with the reverse true for those who said they voted for opposition parties.
National Party voters were less likely to support (58 percent) but also less likely to oppose the bill (8 percent) than Act (73 percent and 12 percent) or New Zealand First (71 percent and 18 percent) voters.
Of those who voted for opposition parties, Greens supporters were the most in favour - with 24 percent supporting the bill and 49 percent opposed, compared with Labour (18% support and 58 percent oppose), or Te Pāti Māori (3 percent support and 53 percent oppose) supporters.
The scientific poll of 1000 eligible voters aged over 18 was taken between 4 and 6 June, with 800 contacted by phone and 200 by online panel, with a +/- 3.1 percent margin of error at the 95 percent confidence level. Results were weighted for gender, age and location.
